We now discuss what the existence of the transformation
(REF ) implies at the level of the consistency around the
cube.
To make the discussion easier to follow, here will work with
the equation on the single cell (REF ).
We will show
that allowing a simultaneous transformation of all the points
\(x\) , \(x_{1}\) , \(x_{2}\) , \(x_{3}\) , \(x_{12}\) , \(x_{13}\) , \(x_{23}\) 
and \(x_{123}\) , i.e., by using the group \(\operatorname{M\ddot{o}b}^{8}\) 
it is possible to produce a sextuplet of consistent quad-equations
of which (REF ) is the base equation.
Indeed let us start from the sextuple of equations for the
(REF ) equation given in [1]}
\(A = \frac{x+e_{2}}{x+e_{1}}\frac{x_{12}+o_{2}}{x_{12}+o_{1}} -\frac{x_{1}+e_{2}}{x_{1}+o_{1}}\frac{x_{2}+o_{2}}{x_{2}+e_{1}},  \\\bar{A} = \frac{x_{3}+e_{2}}{x_{3}+e_{13}}\frac{x_{123}+o_{2}}{x_{123}+o_{1}} -\frac{x_{13}+e_{2}}{x_{1}+o_{1}}\frac{x_{23}+o_{2}}{x_{23}+e_{1}},  \\B = \frac{x+e_{3}}{x+e_{2}}\frac{x_{23}+o_{3}}{x_{23}+o_{2}} -\frac{x_{2}+e_{3}}{x_{2}+o_{2}}\frac{x_{3}+o_{3}}{x_{3}+e_{2}},  \\\bar{B} = \frac{x_{1}+e_{3}}{x_{1}+e_{2}}\frac{x_{123}+o_{3}}{x_{123}+o_{2}} -\frac{x_{2}+e_{3}}{x_{12}+o_{2}}\frac{x_{13}+o_{3}}{x_{13}+e_{2}},  \\C = \frac{x+e_{3}}{x+e_{1}}\frac{x_{13}+o_{3}}{x_{13}+o_{1}} -\frac{x_{1}+e_{3}}{x_{1}+o_{1}}\frac{x_{3}+o_{3}}{x_{3}+e_{1}},  \\\bar{C} = \frac{x_{2}+e_{3}}{x_{2}+e_{1}}\frac{x_{123}+o_{3}}{x_{123}+o_{1}} -\frac{x_{12}+e_{3}}{x_{12}+o_{1}}\frac{x_{23}+o_{3}}{x_{23}+e_{1}}. \) 
